Chaos Emeralds can be found in the Special Zones. To get to the Special Zone you need to finish any Act (that doesn't have a boss) with 50 rings or more. Once you get to the end of the act there will be a large ring, jump into that and you will be taken to the Special Zone. This is like a rotating maze, so avoid the "Goal" circles as that will transport you out of the zone, and make your way to the emerald, they're usually encased in diamonds that you need to roll/jump on to make them disappear. Once you collect the Chaos Emerald, the achievement is yours.